Question

If A-B\[ = \left[ {\begin{array}{*{20}{c}}
  1&5 \\
  3&7
\end{array}} \right]\],\[2A - 3B = \left[ {\begin{array}{*{20}{c}}
  { - 2}&5 \\
  0&7
\end{array}} \right]\]then matrix B is equal to
A.\[\left[ {\begin{array}{*{20}{c}}
  { - 4}&{ - 5} \\
  { - 6}&{ - 7}
\end{array}} \right]\]
B. \[\left[ {\begin{array}{*{20}{c}}
  0&6 \\
  { - 3}&7
\end{array}} \right]\]
C. \[\left[ {\begin{array}{*{20}{c}}
  2&{ - 1} \\
  { - 6}&{ - 7}
\end{array}} \right]\]
D. \[\left[ {\begin{array}{*{20}{c}}
  6&{ - 1} \\
  0&1
\end{array}} \right]\]

Answer 1

Hint: We use matrix subtraction to subtract the two given equations. Again subtract the obtained equation and one of the equations from the equations given in the question which helps us to cancel one of the matrices.
* Matrix addition refers to adding the respective term of a matrix to the corresponding term of another matrix having the same order.
* Order of a matrix having ‘m’ rows and ‘n’ columns is given by \[m \times n\]

Complete step by step answer:
We are given two equations
We can clearly see the order of the matrices in both equations is \[2 \times 2\]
\[A - 2B = \left[ {\begin{array}{*{20}{c}}
  1&5 \\
  3&7
\end{array}} \right]\] ………………….… (1)
\[2A - 3B = \left[ {\begin{array}{*{20}{c}}
  { - 2}&5 \\
  0&7
\end{array}} \right]\] ………………...… (2)
Subtract equation (1) from equation (2)
Subtract LHS of equation (1) from LHS of equation (2) and subtract RHS of equation (1) from RHS of equation (2)
\[ \Rightarrow (2A - 3B) - (A - 2B) = \left[ {\begin{array}{*{20}{c}}
  { - 2}&5 \\
  0&7
\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}
  1&5 \\
  3&7
\end{array}} \right]\]
Calculate LHS by opening brackets and RHS by subtracting corresponding terms of matrices
\[ \Rightarrow 2A - 3B - A + 2B = \left[ {\begin{array}{*{20}{c}}
  { - 2 - 1}&{5 - 5} \\
  {0 - 3}&{7 - 7}
\end{array}} \right]\]
Collect the same variables in LHS of the equation
\[ \Rightarrow (2A - A) + (2B - 3B) = \left[ {\begin{array}{*{20}{c}}
  { - 3}&0 \\
  { - 3}&0
\end{array}} \right]\]
Calculate the values inside the bracket in LHS of the equation
\[ \Rightarrow A - B = \left[ {\begin{array}{*{20}{c}}
  { - 3}&0 \\
  { - 3}&0
\end{array}} \right]\] ………………...… (3)
Now we subtract equation (1) from equation (3)
\[ \Rightarrow (A - B) - (A - 2B) = \left[ {\begin{array}{*{20}{c}}
  { - 3}&0 \\
  { - 3}&0
\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}
  1&5 \\
  3&7
\end{array}} \right]\]
Calculate LHS by opening brackets and RHS by subtracting corresponding terms of matrices
\[ \Rightarrow A - B - A + 2B = \left[ {\begin{array}{*{20}{c}}
  { - 3 - 1}&{0 - 5} \\
  { - 3 - 3}&{0 - 7}
\end{array}} \right]\]
Collect the same variables in LHS of the equation
\[ \Rightarrow (A - A) + (2B - B) = \left[ {\begin{array}{*{20}{c}}
  { - 4}&{ - 5} \\
  { - 6}&{ - 7}
\end{array}} \right]\]
Calculate the values inside the bracket in LHS of the equation
\[ \Rightarrow B = \left[ {\begin{array}{*{20}{c}}
  { - 4}&{ - 5} \\
  { - 6}&{ - 7}
\end{array}} \right]\]
Therefore, the value of matrix B is\[\left[ {\begin{array}{*{20}{c}}
  { - 4}&{ - 5} \\
  { - 6}&{ - 7}
\end{array}} \right]\]

Hence, the correct option is A.

Note:
Students are likely to make mistakes while subtracting the terms of the matrix, keep in mind the corresponding terms of the matrices are to be added or subtracted. Also, check the order of matrices is same or not before applying addition or subtraction as an operation.